5 Network of Detectors

Several gravitational-wave detectors can observe gravitational waves from the same source. For example a network of bar detectors can observe a gravitational-wave burst from the same supernova explosion, or a network of laser interferometers can detect the inspiral of the same compact binary system. The space-borne LISA detector can be considered as a network of three detectors that can make three independent measurements of the same gravitational-wave signal. Simultaneous observations are also possible among different types of detectors. For example, a search for supernova bursts can be performed simultaneously by resonant and interferometric detectors [21].

Let us consider a network consisting of N gravitational-wave detectors and let us denote by xI the data collected by the Ith detector (I = 1,...,N). We assume that noises in all detectors are additive, so the data xI is a sum of the noise nI in the Ith detector and eventually a gravitational-wave signal h I registered by the Ith detector,

xI(t) = nI(t) + hI(t), I = 1, ...,N. (127 )
It is convenient to collect all the data streams, all the noises, and all the gravitational-wave signals into column N × 1 matrices denoted respectively by x, n, and h,
( ) ( ) ( ) x1(t) n1(t) h1(t) | . | | . | | . | x(t) := ( .. ) , n(t) := ( .. ) , h(t) := ( .. ) , (128 ) xN(t) nN(t) hN(t)
then Eqs. (127View Equation) can shortly be written as
x(t) = n(t) + h(t). (129 )
If additionally all detectors’ noises are stationary, Gaussian, and continuous-in-time random processes with zero means, the network log likelihood function is given by
logΛ [x] = (x |h ) − 1(h|h), (130 ) 2
where the scalar product ( ⋅ | ⋅ ) is defined by
∫ ∞ T − 1 ∗ (x |y) := 4ℜ &tidle;x(f) ⋅ Sn(f ) ⋅ &tidle;y(f )df. (131 ) 0
Here Sn is the one-sided cross spectral density matrix of the noises of the detector network, which is defined by (here E denotes the expectation value)
[ ∗ ′T] 1 ′ E &tidle;n(f ) ⋅ &tidle;n (f ) = --δ(f − f )Sn(|f|). (132 ) 2

The analysis is greatly simplified if the cross spectrum matrix Sn is diagonal. This means that the noises in various detectors are uncorrelated. This is the case when the detectors of the network are in widely separated locations, like, for example, the two LIGO detectors. However, this assumption is not always satisfied. An important case is the LISA detector where the noises of the three independent responses are correlated. Nevertheless for the case of LISA one can find a set of three combinations for which the noises are uncorrelated [108, 112]. When the cross spectrum matrix is diagonal the network log likelihood function is just the sum of the log likelihood functions for each detector.

Derivation of the likelihood function for an arbitrary network of detectors can be found in [49Jump To The Next Citation Point]. Applications of optimal filtering for observations of gravitational-wave signals from coalescing binaries by networks of ground-based detectors are given in [63, 41, 62, 103], and for the case of stellar-mass binaries observed by LISA space-borne detector in [78, 118]. The single-detector ℱ-statistic for nearly monochromatic gravitational waves from spinning neutron stars was generalized to the case of a network of detectors (also with time-varying noise curves) in [42] (in this work the ℱ-statistic was also generalized from the usual single-source case to the case of a collection of known sources). The reduced Fisher matrix [defined in Eq. (81View Equation)] for the case of a network of interferometers observing spinning neutron stars has been derived and studied in [110].

Network searches for gravitational-wave burst signals of unknown shape are often based on maximization of the network likelihood function over each sample of the unknown polarization waveforms h+ and h× and over sky positions of the source [52, 95]. A least-squares-fit solution for the estimation of the sky location of the source and the polarization waveforms by a network of three detectors for the case of a broadband burst was obtained in [56].

There is also another important method for analyzing the data from a network of detectors – the search for coincidences of events among detectors. This analysis is particularly important when we search for supernova bursts, the waveforms of which are not very well known. Such signals can be easily mimicked by the non-Gaussian behavior of the detector noise. The idea is to filter the data optimally in each of the detectors and obtain candidate events. Then one compares parameters of candidate events, like, for example, times of arrivals of the bursts, among the detectors in the network. This method is widely used in the search for supernovae by networks of bar detectors [24]. A new geometric coincident algorithm of combining the data from a network of detectors was proposed in [117]. This algorithm employs the covariances between signal’s parameters in such a way that it associates with each candidate event an ellipsoidal region in parameter space defined by the covariance matrix. Events from different detectors are deemed to be in coincidence if their ellipsoids have a nonzero overlap. The coincidence and the coherent strategies of multidetector detection of gravitational-wave signals from inspiralling compact binaries have been compared in [96Jump To The Next Citation Point, 97Jump To The Next Citation Point, 32Jump To The Next Citation Point].  [96] considered detectors in pairs located in the same site and [97] pairs of detectors at geographically separated sites. The case of three detectors (like the network of two LIGO detectors and the Virgo detector) has been considered in detail in  [32], where it was demonstrated that the hierarchical coherent pipeline on Gaussian data has a better performance than the pipeline with just the coincident stage.

A general framework for studying the effectiveness of networks of interferometric gravitational-wave detectors has been proposed in [125]. Using this framework it was shown that adding a fourth detector to the existing network of LIGO/VIRGO detectors can dramatically increase, by a factor of 2 to 4, the detected event rate by allowing coherent data analysis to reduce the spurious instrumental coincident background.

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